# A Haptic Model for the Quantum Phase of Fermions and Bosons in Hilbert Space Based on Knot Theory

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## Abstract

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## 1. Introduction

## 2. Great Circles and Double Windings in the $\left(\mathbf{2}\mathit{\pi}\right)$-Realm

## 3. Jones Polynomials of Bosonic and Fermionic States in the $\left(\mathbf{4}\mathit{\pi}\right)$-Realm

## 4. A Generalization of the “Dirac Belt” Trick in the $\left(\mathbf{4}\mathit{\pi}\right)$-Realm

**1**) = $2j+1$ (in terms of the coordinate $z\in C{P}^{1}$, this rotation is described by a Möbius transformation matrix ${U}_{q}$).

## 5. Relation Between Nodes, Twists and Knots

## 6. Observable Effects of the Möbius-Strip Topology

## 7. Summary and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The paper strip model for the phase factor $exp[+i(p/2\left)\varphi \right]$: An odd number of twists $\left(\right|p|=1,3)$ leads to a Möbius strip topology of the paper strip with only one surface. On a Möbius strip, after a rotation of $2\pi $, the original position is reached upside-down. The modes with an even number of twists $\left(\right|p|=0,2,4)$ have two surfaces. A rotation of $2\pi $ leads to the original position on the paper strip. Right-moving twists (R) emerge for $p>0$, while left-moving twists (L) have $p<0$.

**Figure 2.**On the left: Derivation of the recursive formula for the Jones polynomials for the highest weight states $|j,\pm j\rangle $. The general spin states $j=\{l-1/2,\phantom{\rule{4pt}{0ex}}l,\phantom{\rule{4pt}{0ex}}l+1/2\}$ (with integer spin l) are obtained from the braid with $2l$ twists by insertion of one of the three boxes, respectively. On the right: Paper strips with $p=1,2,3$ twists (R, RR, RRR) embedded in a torus, corresponding to spin $j=1/2,1,3/2$, respectively. The red and blue edges of the paper strip correspond to the braids.

**Figure 3.**In the $\left(4\pi \right)$-realm of $SU\left(2\right)$, the paper strip describing the quantum phase has a length of $\left(4\pi \right)$ and always has two surfaces. In other words, in contrast to the $\left(2\pi \right)$-realm, no double-valuedness of the wave function can emerge. Inner twists can be performed in three different “directions”, which can be described using the quaternions $I\equiv -i{\sigma}_{1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}J\equiv -i{\sigma}_{2}$ and $K\equiv -i{\sigma}_{3}$. The non-commutativity is reproduced by the motion of the paper strip in the $\left(4\pi \right)$-realm shown here, since the order of operations matters for the result. In particular, $IJK|*\rangle =-1|*\rangle $ for an arbitrary initial state $|*\rangle $.

**Figure 4.**The operation ${I}^{2}={I}^{-2}=-1$ introduces a single loop in the trivial state. The loop is equivalent to two clockwise twists $(++)$ or two counter-clockwise twists $(--)$. Note that twists in the $\left(4\pi \right)$-realm (denoted ±) must carefully be distinguished from those in the $\left(2\pi \right)$-realm (denoted R/L).

**Figure 5.**

**Upper row**: Cutting the Möbius band $j=1/2$ (R in Figure 1) in the middle leads to a Dirac belt with four twists ${T}^{\prime}=4$, denoted by $(++++)$. Due to this operation, the length $2\pi $ of the paper strip is extended to $4\pi $, and the Möbius band-topology disappears.

**Lower row**: The same operation for $j=1$ (RR in Figure 1) leads to two intertwined, identical copies of the original state (Hopf link), each again with two twists ${T}^{\prime}=T\times T$, in this case $(++)\times (++)$. The phase $2\pi $ is not extended to $4\pi $.

**Figure 6.**The paper strip visualization for spin $j=1/2,\dots 3$ in the full Hilbert space, which is obtained from the model in the $\left(2\pi \right)$-realm shown in Figure 1 by cutting the strip once lengthwise in the middle to double the phase. For fermions, a generalized Dirac belt emerges, described by a single knot. For bosons, two identical, intertwined copies of the original state shown in Figure 1 are obtained, leading to a generalized Hopf-link. Note that this is only possible for knots with inner twists ${T}^{\prime}=4j+2$ or links with inner twists ${T}^{\prime}=2j\times 2j$ (here, $j=l$ must be integer). After Hopf-mapping, these twists are mapped to $2j$ nodes in the stellar representation; see Figure 7.

**Figure 7.**Relation between the stellar representation in the $\left(2\pi \right)$-realm and of the the knot structure in the $\left(4\pi \right)$-realm, illustrated for $j=1/2,\phantom{\rule{4pt}{0ex}}j=1$ and $j=3/2$. Note that the number $p=2j$ of nodes (sometimes called “stars”) in the stellar representation on ${S}_{2}$ (corresponding to the number of twists Tin the paper strip model in the $\left(2\pi \right)$-realm; see Figure 1) determines the knot structure in the $\left(4\pi \right)$-realm. The mapping h is the Hopf mapping (${h}^{-1}$ being the inverse Hopf mapping). In Figure 6, the knot structure in the $\left(4\pi \right)$-realm is represented in the paper strip model. The two-to-one Hopf mapping is achieved in the paper strip model by gluing the strips together, as shown in Figure 5.

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**MDPI and ACS Style**

Heusler, S.; Ubben, M.
A Haptic Model for the Quantum Phase of Fermions and Bosons in Hilbert Space Based on Knot Theory. *Symmetry* **2019**, *11*, 426.
https://doi.org/10.3390/sym11030426

**AMA Style**

Heusler S, Ubben M.
A Haptic Model for the Quantum Phase of Fermions and Bosons in Hilbert Space Based on Knot Theory. *Symmetry*. 2019; 11(3):426.
https://doi.org/10.3390/sym11030426

**Chicago/Turabian Style**

Heusler, Stefan, and Malte Ubben.
2019. "A Haptic Model for the Quantum Phase of Fermions and Bosons in Hilbert Space Based on Knot Theory" *Symmetry* 11, no. 3: 426.
https://doi.org/10.3390/sym11030426